In simple terms, how do you find the domain and range of a function?

Hello!

Let's start from the domain. Usually this word is used in the sense "all values of `x` which are possible to use for the given formula". There are some things which is impossible: division by zero, finding square root of the negative number, finding logarithm of non-positive number and so on.

So when we see a formula, we have to analyse whether it contains division, roots and so on. Example:

`f(x)=sqrt(x+2)/x.`

There is `x`...

Hello!

Let's start from the domain. Usually this word is used in the sense "all values of `x` which are possible to use for the given formula". There are some things which is impossible: division by zero, finding square root of the negative number, finding logarithm of non-positive number and so on.

So when we see a formula, we have to analyse whether it contains division, roots and so on. Example:

`f(x)=sqrt(x+2)/x.`

There is `x` in the denominator, so `x!=0.` Also, there is a square root, so `x+2gt=0` or `xgt=-2.` The resulting domain is `[-2,0) uu (0,+oo).`

Sometimes finding the domain may be difficult, but the idea is as above. Try the function `g(x)=sqrt(2-sqrt(3-x))` (isn't very difficult).

The range of a function is the set of its values. To decide whether some `y` is in range of `f(x)` we have to consider the equation `f(x)=y` for `x.` If at least one solution exists, then `y` is in the range.

For example, the range of a linear function `f(x) =ax+b` is the set of all real numbers if `a!=0,` and the set of the only one element `{b}` if `a=0.` Another example: you probably know that the function `g(x)=sin(x)` takes all values in `[-1,1]` but no values outside; this means that the range of `sin(x)` is `[-1,1].`

Finding a range may also be difficult.

Note that sometimes it is useful to restrict a domain "manually". For example, one may consider the function `h(x)=x^2+1` on the segment `[1,2]` only. Then its domain is `[1,2]` and the range is `[2,5]`, while for the "unrestricted" function the domain is all real numbers and the range is `[1,+oo).`

Discuss writing a paper that has validity. Cover fact and fiction, supporting evidence, bias, and logic.

A paper that has validity must be credible, which means convincing and believable. The key to achieving this lies in backing up or illustrating your ideas with quotes and examples (supporting evidence). For example, if you want to prove in an English paper that a character in a novel is evil, you need to go back to the text and find quotes or examples that show this character's evil nature. Likewise, if you are asked to research and write about ways the food industry encourages obesity, you have to find specific examples of what the food industry does. It's not enough to say that food conglomerates stuff calories into junk food. Instead, you have to find examples of specific types of junk food and discuss how and why they are so caloric. In short, to write a valid paper, you must back up every statement you make with a quote or evidence that convinces your reader that what you say is true or at least convincing enough to be worth considering.

To produce a valid or persuasive paper, you also need to avoid biased sources and to be aware of your own possible biases. If you are biased, you might throw out evidence that contradicts what you already think you know and not offer a balanced argument. This is poor scholarship. A valid paper admits the strengths of its opponents and then argues against them. It does not include information that is not factual or that has been distorted in some way to make a point. For example, if you were quoting a source, you would not ellipse (leave out) words that were important to understanding what the author really meant. You would also use sources that any educated person would consider authoritative and unbiased, such as The New York Times or a peer-reviewed scholarly journal.

A good paper is logically coherent, meaning the evidence you have gathered actually supports the points you are trying to make. In a valid paper, you point out any logical inconsistencies  (fallacies) in the arguments your sources make. For example, one of your sources might attack a person rather than an idea, which is called making an ad hominem argument. A classic example of this would be to say we should not build a strong highway system because Hitler did. Hitler was clearly a terrible person, but the German highway system was one of his few good ideas. 

Finally, we come to fact and fiction. You clearly want your papers to contain fact, not fiction, but this speaks as well to another issue. A convincing paper contains both logos (facts and statistics) and pathos (true stories). Very few people will read a paper this is simply a pile of facts, such as a list of average temperatures over the past 60 years. Statistics alone don't make an argument. Every paper needs true stories that illustrate what their facts mean in people's lives. At the same time, you don't want your paper to be all story: this is like serving a meal of nothing but dessert. A reader will trust much more in your paper's validity only if it includes ample servings of  the "protein and vitamins" that facts provide. 

In short, a valid paper uses believable sources, uses many quotes and examples to back up its ideas, balances facts with true stories to illustrate its points, works to use unbiased sources and points out the logical fallacies it finds in arguments. 

What do lines 81-85 of Beowulf and the beginning stanza of Sir Gawain and the Green Knight have in common?

Lines 81-85 of Beowulf finalize the initial description of Heorot and allude to the future destruction of the estate in a fiery inferno. Likewise, the first stanza of Sir Gawain and the Green Knight chronicles the burning of Troy and connects the far-flung Trojans with the eventual settling of Britain. In a nutshell, both passages deal with the destruction of mythological cities and strongholds. The major difference between the two is that, while Beowulf looks...

Lines 81-85 of Beowulf finalize the initial description of Heorot and allude to the future destruction of the estate in a fiery inferno. Likewise, the first stanza of Sir Gawain and the Green Knight chronicles the burning of Troy and connects the far-flung Trojans with the eventual settling of Britain. In a nutshell, both passages deal with the destruction of mythological cities and strongholds. The major difference between the two is that, while Beowulf looks ahead toward a future cataclysm, Gawain delves into the past to recount a mythological event. It's also worth noting that both poems are keen to connect the events of their respective narratives with a place of cultural and/ or mythological significance. By doing so, the poet is able to root or connect the poem to a location that would have been familiar to most audiences, making it easier to connect to story.    

What is the Doppler effect? (write in kid language)

Doppler effect is the change in the apparent frequency of a source when relative motion takes place between the source and the observer. Think of what happens when you are standing right next to the road and an ambulance is approaching. As the ambulance gets nearer and nearer, you can hear its siren at successively higher pitch. However, when the ambulance passes you and is moving away from you, the pitch of its siren drops...

Doppler effect is the change in the apparent frequency of a source when relative motion takes place between the source and the observer. Think of what happens when you are standing right next to the road and an ambulance is approaching. As the ambulance gets nearer and nearer, you can hear its siren at successively higher pitch. However, when the ambulance passes you and is moving away from you, the pitch of its siren drops successively. Note that the siren of the ambulance is emitting sound waves at the same rate this whole time. The same can be experienced at a train station. What really happens is, as the source approaches you, the waves are getting nearer and nearer and hence the pitch increases. As the source recedes, the waves are getting farther apart and hence the pitch falls. 

The same is observed if the source is stationary and the observer moves towards it or away from it. Interestingly, the Doppler effect was used in astronomy to determine that galaxies are moving away from us (and each other) and the fact that universe is expanding.

Hope this helps. 

What is the narration of this story? Is it first person retrospective?

First person narratives always employ the words "I," "me," "mine," "my," etc. Edgar Allan Poe's famous story "The Cask of Amontillado" is a good example of first-person retrospective narration. "To Build a Fire" could be told in the first person, but it isn't. The narrative form is third-person with an anonymous and seemingly omniscient narrator. The narrator knows all about what is going on inside the man and even what is going on inside the dog. This anonymous narrator always refers to the man as "he," which is what makes it a third-person narrative. The narrator also knows all about the climate conditions in the Klondike and about such things as the ancestry of the dog. 

Vladimir Nabokov quotes a little rhyming couplet which he offers as a rule for first-person narration. 

The "I" in the story
Cannot die in the story.

If the "I" telling the story were to die before the end of the story, the story could not go on because the narrator would be dead. In "To Build a Fire," the protagonist dies before the story is over. The narrator tells the rest of the story from the perspective of the dog. 

Later, the dog whined loudly. And still later it crept close to the man and caught the scent of death. This made the animal bristle and back away. A little longer it delayed, howling under the stars that leaped and danced and shone brightly in the cold sky. Then it turned and trotted up the trail in the direction of the camp it knew, where were the other food-providers and fire-providers.

This is another indication that this cannot be first-person retrospective, or any other kind of first-person narration. The story is told in the third person and mainly limited to the points of view of the man and the dog, although the author Jack London interjects some straight prose exposition about such things as the setting and the weather conditions. 

In their excellent anthology titled Points of View: An Anthology of Short Stories (Revised edition 1995), the editors, James Moffett and Kenneth R. McElheny, have arranged the stories according to the narrative points of view, starting with the most subjective (e.g., "I Stand Here Ironing") and moving up to the most objective (e.g., "The Lottery"). "To Build a Fire" is not included in Moffett and McElheny's collection, but they would classify it as ANONYMOUS NARRATION--DUAL CHARACTER POINT OF VIEW. The dual characters are, of course, the man and the dog. 

is honors biology hard? What does it cover? I am going to be a freshman in HS in 2016-17. Will it be better to take honors courses and get B's...

The University of California website states that one of the factors considered in admission is “UC-approved” honors, AP and IB courses and that extra weight is assigned to grades received in honors courses. You should check with the counselor of the high school you’ll be attending to find out if the honors biology course meets this requirement. UC also considers “academic performance related to opportunities available in your high school" which suggests that taking advantage of rigorous courses will improve your standing.

You will also need to ask a counselor about enrolling in honors courses as a 10th grader if you don’t do so in 9th grade. However, I think that starting out in honors courses will be beneficial if you plan to enroll at some point in high school. Students in their second year of honors courses will have already learned the expectations and adjusted to the rigor.

Biology is the study of life. You’ll likely be learning about plants, animals, cells, genetics and ecology. It’s likely that the curriculum for honors biology is similar to or the same as the regular biology curriculum. An honors section of a course usually involves learning concepts in more depth. You’ll probably be expected to spend more time outside of class on assignments for honors biology. Honors courses are designed to help prepare students for college, so some independence and self-motivation will be expected. Does your high school have an orientation day for freshmen in advance of registration? If so, this would be a good opportunity to meet the biology teachers and talk to them about the courses offered.

The benefit of taking honors courses in high school extends beyond college admission. You’ll be better prepared for the rigor of college coursework if you challenge yourself in high school. You’ll have the opportunity to learn study skills and time management. Do you know any college students who are graduates of your high school? If so, contact one and ask about his or her high school experience. Students who take challenging courses in high school usually have an easier time in college and are appreciative, although belatedly, of the hard work that was required of them in high school.

I know this doesn’t directly answer your question as I can’t speak for colleges, but I hope it gives you some direction in making your decision. Don’t forget to talk to your parents about your opportunities and choices.

If macbeth were a modern man, who would he be? What do these men have in common with each other?

A good modern parallel to Macbeth might be presidential candidate Ted Cruz. What both men have in common is boundless ambition, a willingness to do almost anything to get to the top. Now, Cruz is NOT willing to murder others to clear his path! But without going to those lengths, he is very ambitious. We should also note that Macbeth's wife pushed him to commit murder--Cruz's wife doesn't share Lady Macbeth's murderous tendencies at all.

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A good modern parallel to Macbeth might be presidential candidate Ted Cruz. What both men have in common is boundless ambition, a willingness to do almost anything to get to the top. Now, Cruz is NOT willing to murder others to clear his path! But without going to those lengths, he is very ambitious. We should also note that Macbeth's wife pushed him to commit murder--Cruz's wife doesn't share Lady Macbeth's murderous tendencies at all.

Let's focus on ambition, leaving aside the issue of murder. Again, Cruz is not a murderer.

But as a recent op-ed piece in the New York Times (see link below) notes, Cruz wants to be the top dog, just as Macbeth did. 

Here is Bruni on Cruz:

"It’s striking how many explorations of his past wind up focusing on the magnitude of his confidence, the scale of his ambitions and the off-putting nakedness of both.

As a cocky teenager, he said that his life goals were to “take over the world, world domination, you know, rule everything.” He separately wrote of plans to “achieve a strong enough reputation and track record to run for — and win — president of the United States.”

We know that the whole thrust of Macbeth is his lust to become king. Macbeth's theme, however, underscores the folly of ambition without mercy, compassion or moderation. Naked ambition, Shakespeare says, leads to disaster. Macbeth functions as a warning or cautionary tale. Should we send Mr. Cruz a copy of Shakespeare's play?